Free-surface waves generation by a fully submerged wake

Abstract

The nonlinear evolution of two-dimensional instability waves in a fully submerged wake is studied numerically through direct numerical simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions on the ocean surface. For a parallel, fully submerged wake flow, the sinuous mode of linear instability is more unstable than the varicose mode. Therefore, the nonlinear evolution of the instability results in a staggered-vortex pattern in the bulk of the fluid, while the free-surface signature depends on the submergence depth of the mean velocity profile and the Froude number of the flow. Specifically, for large submergence depth and low Froude number, the flow reaches a quasi-equilibrium state, where the free surface takes the form of a propagating gravity wave with a very small height. However, for the same submergence depth, increasing the Froude number beyond a certain value causes breaking of the free-surface wave. For high Froude number, wave breaking is caused by the presence of a sharp vertical velocity shear along the free surface for deep and shallow wakes alike. For small submergence depth, on the other hand, the free-surface wave breaks even for low Froude number, because of the sharp horizontal velocity shear that is induced along the free surface by the vortices of the flow.